This study examines a non-linear Cramér-Lundberg risk model, in order to determine the adjustment coefficient (r) when the claims have Weibull distribution. Similar insurance models have been studied in the literature, usually when premiums are linear. However, in some real-world problems, the increase in revenue may not be linear. In such cases, it is significant to consider nonlinear risk models. Accordingly, in this study, a nonlinear Cramér-Lundberg risk model is mathematically constructed and investigated when the premium function is \( p(t)=c\sqrt{t} \) . As is known, the adjustment coefficient plays a substantial role in evaluating the ruin probability. Thus, a detailed examination of this coefficient is important. However, it is a challenging process to derive the exact value of r from an integral equation when the claims have Weibull distribution. For this reason, the adjustment coefficient is investigated in this study by using approximation methods and a practical algorithm proposed to calculate approximate result with the desired closeness to the true value.